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ADC801S - ADVANCED CALCULUS THEORY - 2ND OPP - JULY 2022 |
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NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH, APPLIED SCIENCES AND NATURAL RESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION:
QUALIFICATION
COURSE CODE:
SESSION:
DURATION:
Bachelor
CODE:
of Science (Hons)
08BSHM
ADC8015
JULY 2022
3 HOURS
in Applied
LEVEL:
COURSE
PAPER:
MARKS:
Mathematics
8
NAME: ADVANCED
THEORY
100
CALCULUS
SUPPLEMENTARY
EXAMINER:
MODERATOR:
/ SECOND
OPPORTUNITY EXAMINATION
DR. DSI ITYAMBO
PROF. OD MAKINDE
QUESTION
PAPER
INSTRUCTIONS
1. Attempt all the questions in the booklet provided.
2. Show clearly all the steps used in the calculations.
3. All written work must be done in black or blue inked, and sketches must be done in pencil.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 2 PAGES (Including this front page)
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Question 1.
Consider the equation PV = knT’, where k and n are constants. Show that
av OP OP _
OT OP AV
[10]
Question 2.
Find the local extreme values and the saddle points of the function f(z, y) = x? + 2xy + 3y?.
[12]
Question 3.
Use the method of Lagrange multipliers to find the minimum and maximum values of the function
f(a, y) = 2x? + y? + 2, where x and y lie on the ellipse C' given by x? + 4y? -4=0.
[15]
Question 4.
Let F = (2az + y)i+ 2ayj + (x? + 32?)k.
a) Determine whether F is a conservative vector field. If it is, find a potential function for F.
b) Evaluate {,F - dr, where C is the curve given by r(¢) = t7i + (¢ + 1)j + (2¢ — 1)k, where
O0<t<l.
[19,7]
Question 5.
Evaluate {., xyz? dS, where C is the line segment joining (—1,—3, 0) to (1, —2, 2)
[10]
Question 6.
Let f be a differentiable function of x, y and z, and let F(z,y,z) = P(z,y,z)it+ Q(2z,y,z)j+
R(x, y,z)k, where P, Q and R are differentiable functions of x, y and z. Prove that
div(fF) = fdivF +F. Vf.
[10]
Question 7.
Use Green’s Theorem to evaluate ¢ (3y — eS™*) da — (7a + /y* + 1) dy, where C is the circle
of radius 9 centred at the origin. Cc
[9]
Question 8.
Evaluate the integral /f/ 8xyz dV over the box B = [2,3] x [1,2] x [0, 1].
[8]